NASA Contractor Report 195342

/iJ4_

Efficient and Accurate Explicit Integration

Algorithms With Application toViscoplastic Models

Vinod K. Arya

University of Toledo

Toledo, Ohio

August 1994

(NASA-CR-195342) EFFTCTENT AND

ACCURATE EXPLTCIT INTEGRATION

ALGORITHMS WITH APPLICATION TO

VISCOPLASTIC MODELS Final Report

(To|edo Univ.) 25 p

G3/64

N95-11464

Unclas

0022319

Prepared for

Lewis Research Center

Under Cooperative Agreement NCC3-120

National Aeronautics and

Space Administration

https://ntrs.nasa.gov/search.jsp?R=19950005051 2020-05-21T21:55:35+00:00Z

EFFICIENT AND ACCURATE EXPLICIT INTEGRATION

ALGORITHMS WITH APPLICATION TO

VISCOPLASTIC MODELS

Vinod K. Arya*

University of Toledo

Toledo, Ohio

SUMMARY

Several explicit integration algorithms with self-adaptive time integration strategies are developed

and investigated for efficiency and accuracy. These algorithms involve the Runge-Kutta second order, the

lower Runge-Kutta method of orders one and two, and the exponential integration method. The algorithms are

applied to viscoplastic models put forth by Freed and Verrilli and Bodner and Partom for thermal/mechanical

loadings (including tensile, relaxation, and cyclic loadings).

The large amount of computations performed showed that, for comparable accuracy, the efficiency of

an integration algorithm depends significantly on the type of application (loading). However, in general, for

the aforementioned loadings and viscoplastic models, the exponential integration algorithm with the proposed

self-adaptive time integration strategy worked more (or comparably) efficiently and accurately than the other

integration algorithms. Using this strategy for integrating viscoplastic models may lead to considerable saving

in computer time (better efficiency) without adversely affecting the accuracy of the results. This conclusion

should encourage the utilization of viscoplastic models in the stress analysis and design of structural

components.

INTRODUCTION

The need for an accurate assessment of the time-dependent, inelastic response of structural

components has led to the development of numerous constitutive material models, called viscoplastic models.

Accurate assessment of inelastic response is an essential input for predicting the service life of components.

Viscoplastic models incorporate all aspects of inelastic deformation, including plasticity, creep, andrelaxation. These models do not endeavor to artificially split the inelastic strain into plastic and creep

components. Alternatively, they treat all the inelastic strain (including plasticity, creep, and relaxation) as a

single time-dependent entity. This "unified" representation of time-dependent, inelastic strain enables these

models to automatically include the observed interactions at high temperatures among plasticity, creep, andrelaxation. These models, therefore, provide a more realistic and accurate description and response of the

time-dependent, inelastic behavior of materials. The input of inelastic strain, calculated by using these

viscoplastic models, into life prediction methods thus leads to a more accurate determination of the service

life of structural components.

The mathematical framework of most viscoplastic models consists of a flow law that relates the

inelastic strain rate to the deviatoric stress and evolution equations for internal state variables that representthe resistance of the material to further inelastic flow. Most models use two internal state variables, a tensor

and a scalar. The tensorial and scalar variables represent the kinematic and isotropic hardenings of the

material, respectively. The form of the evolution laws for the internal state variables is based on the well-

known Bailey-Orowan theory (ref. 1), which states that the high-temperature deformation of a material takes

place under the influence of two competing mechanisms. One mechanism represents the hardening processwith accumulated deformation, and the other represents a thermal recovery or softening process proceeding

* Resident Research Associate at Lewis Research Center.

withtimeat temperature. The evolution rate of internal state variables is then defined as the difference

between the rates of the hardening and recovery processes. Under steady-state conditions these twomechanisms balance each other.

There is a significant incentive to make viscoplastic models more realistic by incorporating into them

as much material science as possible. Consequently, the mathematical structure of viscoplastic models is very

complex. The constitutive equations governing the flow behavior and the evolution of internal state variables

are, in general, highly nonlinear and mathematically "stiff." This inherent stiff character of the constitutive

equations causes major problems during their numerical integration. Present-day structural engineering

problems, such as those encountered in the aerospace and nuclear industries, involve complex thermal/

mechanical loading histories. For the solution of these problems the entire loading history has to be traced,

requiring these equations to be integrated many times. The cost and computer time involved prohibit using the

conventional method of employing small time steps for accurate integration of these stiff equations. There is,

therefore, a strong need to develop efficient and accurate integration algorithms with a self-adaptive, time-step

strategy that can achieve the desired accuracy and stability over the entire integration range (loading

histories).

Keeping the aforementioned requirements in mind, numerous researchers have proposed a number of

integration algorithms with self-adaptive, time-step control. These algorithms are based on explicit or implicit

integration methods. Kurnar et al. (ref. 2) and Banthia and Mukherjee (ref. 3) present several such integration

algorithms and show that for the viscoplastic model due to Hart (ref. 4) using a simple, one-step, explicit

Euler integration scheme with automatic time-step control is advantageous. Gear (ref. 5), Cormeau (ref. 6),

Hughes and Taylor (ref. 7), and Hornberger and Stature (ref. 8) have suggested special implicit integrationschemes. Numerous other attempts to integrate these stiff equations have also been made, and useful

information in this regard is available in the literature (refs. 9 and 10). Walker (refs. 11 and 12) and Freed

et al. (ref. 13) have proposed some explicit and implicit asymptotic exponential integration algorithms and

applied them to the integration of viscoplastic models. (Note that the references mentioned in this paragraph

are by no means exhaustive. They are rather indicative of the importance assigned by various researchers to

the development of efficient and accurate time integration strategies for use with viscoplastic models.)

We have been involved with the development of suitable integration strategies for time integration of

viscoplastic models. A previous work by Arya et al. (ref. 14) reports that a simple, self-adaptive time

integration strategy involving the explicit forward Euler method, although only conditionally stable, works very

successfully if the time step is properly controlled at each integration step. The main advantage of using an

explicit integration scheme is the ease with which it can be implemented. Another favorable aspect of using

explicit integration schemes is that these do not require the assembly and inversion of Jacobian matrices. This

fact is of great advantage and significance especially when the number of equations involved is large. The

advantage of using implicit integration schemes is that they may allow large time steps without affecting the

stability of the integration method. Cormeau (ref. 6) has, however, shown that several implicit methods suffer

from the same time-step restrictions as does an explicit method, and in such circumstances the implicit

integration methods offer no special advantages over the simpler explicit methods.

In light of the aforementioned facts, several explicit time integration strategies are presented in this

paper. Detailed investigations were made to judge their efficiency, accuracy, and suitability for application to

viscoplastic models. The integration strategies were developed and investigated in conjunction with the

following four explicit integration methods:

(1) Runge-Kutta, second-order method (RK2 method)

(2) Lower Runge-Kutta method of order one (or Euler-Cauchy Method) (RK1L method)

(3) Lower Runge-Kutta method of order two (RK2L method)

(4) Exponential integration method (in conjunction with Runge-Kutta second-order method)

(RK2EXP method)

Method1is currently being used by the author for numerically integrating stiff equations of viscoplastic

models. Methods 2 to 4 have shown promise for integrating stiff equations (refs. 15 and 16). However, their

adaptability and applicability for efficiently and accurately integrating the constitutive equations of

viscoplastic models is yet to be explored. The present study endeavored to do this by developing suitable time

integration strategies that employ these methods.

The integration strategies presented herein are general in nature and can be applied to any

viscoplastic mode/. However, two viscoplastic models are used to illustrate the applicability of these

integration strategies. One viscoplastic model is put forth by Freed and Verrilli (ref. 17) and the other by

Bodner and Partom (ref. 18). Although these two viscoplastic models possess identical mathematical

frameworks, their constitutive equations have several significant differences in their forms and in the functions

they involve. These differences, detailed later in the paper, assign different degrees of mathematical

complexity to the models.

First, the mathematical frameworks of the viscoplastic models put forth by Freed-Verrilli and Bodner-

Partom are briefly described. The integration strategies developed and employed in this paper are then

presented in conjunction with the four explicit integration methods listed earlier. Next, procedures to make the

integration strategies self-adaptive are presented. The integration strategies are then applied to both

viscoplastic models for three different types of loadingsptensile, relaxation, and cyclic. Finally, numerical

results are presented and discussed in order to judge the applicability, efficiency, and accuracy of these

integration strategies in conjunction with the viscoplastic models.

VISCOPLASTIC MODELS

Freed-Verrilli Viscoplastic Model

Following the general mathematical structure adopted by most viscoplastic models and as mentioned

earlier, the Freed-Verrilli (F-V) viscoplastic model (ref. 17) also incorporates two internal state variables. One

a tensorial variable called the backstress, accounts for strain-induced or kinematic hardening. The other, called

the drag stress (strength), represents isotropic hardening and is a scalar variable. The model is based on the

assumption that there is no explicit coupling between the dynamic and static recovery terms in an evolutionary

equation for the internal state. This assumption renders a simpler mathematical framework for the model. The

model also employs a small displacement and a small strain formulation.

The total strain rate _ij is assumed to be composed of elastic _eij, inelastic (including plasticity,

and relaxation) _v, and thermal _t., strain rate components and can be written ascreep,_j

• .e .v "t wherei, j=I,2,3 (1)¢ij =eiy + ei: + eij,

In equation (1), the symbol E denotes the strain and the dot over the symbol indicates its derivative with

respect to time t. Superscripts e, v, and t refer to elastic, viscoplastic (inelastic), and thermal components ofthe strain rate.

•e is related to the stress rate (_ij by Hooke's lawFor an isotropic material the elastic strain rate _ij

• e =l+v_.. v •E ij E 'J - "-E (_ kk _ ij

(2)

where E is Young's modulus, v is Poisson's ratio, and _ij is the Kronecker delta. The repeated subscripts in

equation (2) and elsewhere imply summation over their range.

The deviatoric stress Sij is defined by

Sij = Oij - lOkk_ij (3)

The effective stress E.. has the expressionlj

_'ij : Sij - Bij (4)

where Bij is the backstress.

Flow law.--Defining the invariant J2 by

the flow law is written as

J2 = ½ _ijz_j (5)

(6)

The function 0, called the thermal diffusivity function, contains the temperature dependence of the model. It is

defined by

U_ ifT<O.5T mL kr.,t t,2T) '

(7)

Q is the activation energy, k is the Boltzmann constant, T is the absolute temperature, and Tm is the melting

point of the material.

The function Z, called the Zener-Hollomon parameter, is given as

rAF n, if F < 1

Z=laexp[n(F-1)], if F>I(8)

and A and n are material constants. The function F is defined as

(9)

whereD denotes the drag stress.

Evolution laws.--The differential equations that govern the growth of the internal state variables

(backstress Bij and drag stress D) are

, Ivy,/=H Eij--

and

(10)

(11)

where

12 = 0Z (12)

In equations (I0) and (I I), H, h, and L are inelastic material constants. The constant L denotes the limiting

value of the backstress at kinematic saturation (ref. 17). The recovery function r is defined by using thefollowing equations:

0, if D = D Or(G) = R(G), if D > D O(13)

IAG n-l, if G < 1R(G) = [a exp[n(G- I)]/G, ifG>l

(14)

and

LG = _ (15)

S-D

where S and DO are material constants.

For the theory to be thermodynamically admissible, the dissipativity condition must be satisfied. This

results in the following constraint on the function r:.

(16)

where

=½B/:ij (17)

The values of elastic and inelastic material constants appearing in the preceding equations are given in

table I for copper (ref. 17).

Bodner-Partom Viscoplastic Model

The viscoplastic model put forth by Bodner (ref. 19) in 1968 is one of the fu'st unified constitutivemodels that does not require an explicit yield function (or loading/unloading conditions) to govern the inelasticresponse of a material. The mechanistic basis for the model development stems from the work of Gilman(ref. 20) and Johnston and Gilman (ref. 21). In the Bodner-Partom (B-P) model the inelastic strain rate is

considered to be a function of the current value of stress o and an internal state variable that represents a

measure of hardening or resistance to inelastic flow. Although the general mathematical framework of this

viscoplastic model is akin to that adopted by other viscoplastic models, it has several significant differences

from them. For example, in this model the inelastic strain rate is expressed as a function of stress a and not as

a function of overstress, which is defined as the difference of stress _ and the backstress. Also, the measure of

inelastic hardening in this model is taken to be inelastic (plastic) work and not the inelastic strain as is donein other viscoplastie models.

Several modifications to this model have been proposed from time to time. The modifications of themodel advanced by Bodner and Partom (refs. 18 and 22) included large deformations and isotropic hardening,respectively. Stouffer and Bodner (refs. 23 and 24) included the Bauschinger effect associated with loadreversals and the generally anisotropic nature of inelastic hardening. Bodner (ref. 25) extended the model toinclude evolution of damage. The model used in the present paper (from ref. 18) includes isotropic anddirectional hardening, thermal recovery of hardening, and general temperature dependence of inelastic flow.The constitutive equations of the model are given here.

Flow law.--The total strain rate is again assumed to be the sum of elastic, inelastic (viscoplastic),

and thermal strain rate components, and the inelastic strain rate is assumed to have the form of the Prandtl-Reuss flow law. In symbols

• "e "V "t

eij =eij + f-ij + _-ij, where i, j = 1,2, 3 (18)

-v _ij .v =0£ij = ; Ekk (19)

where

Sly = oij - 380_ (20)

In these equations the symbols e, S, o, and 8 and the superscripts e, v, and t have the same meaning as

defined earlier. A repeated index implies summation over its range, and the parameter _ is defined later.

with

Kinetic equation.--The inelastic deformations are governed by the kinetic relation

2 n} (21)

and

lsoso; _2 = nf/j 2 (23)J2 =_

In these equations Z is an internal state variable that represents the resistance of the material to inelastic flow.This state variable in the model is assumed to be composed of isotropic hardening Z t and directional hardening

ZD components. Also, Do denotes the limiting strain rate in shear and n is a material constant.

Isotropic-hardening evolution laws.--The growth or evolution of the isotropic-hardening component is

governed by

ZI =mI(ZI +OtZ3-ZI)(Vp-A1ZII(ZI-Z2)IZ1] rl(24)

where

6_= m2(_l -a)I/Vp sinO (25)

0 = cos-1 (V_/_/ij)or O=cos-l(uijuij) (26)

uo (2s)

with

z_(o)= Zo; % =oo/_.; %(o)=o; a(o)=o (29)

In these equations Z0 is the initial value of the isotropic hardening variable Z t ; Zl denotes the

limiting (maximum) value of ZI; Z 2 is the fully recovered (minimum) value of Z 1 ; and Z 3 denotes the limiting

(maximum) value of Z D- The parameter 0 is a measure of nonproportionality of loading, and a is a loading-

history-dependent variable that provides an additional hardening increment in the evolution equation for Z !.

The coefficient a 1 corresponds to the limiting value of a, and m I and m 2 denote the hardening rate

coefficients of Z/and ZD, respectively. The rate of inelastic (plastic) work is denoted by Wp. The symbols A 1

and rl denote the recovery coefficient and the recovery exponent for 7_]. The second-order symmetric tensor 13g

represents the directional-hardening state variable.

Directional-hardening evolution law.--The evolution equation for the directional-hardening variable

13/j has the same general form as that for isotropic hardening. It, however, has a tensorial character. It is writtenas

with

(30)

z ° = Bijuo; z°(0) = 0; 13ij(0)= 0 (31)

Here m2 denotes the hardening rate coefficient of Z D. The symbols A 2 and r2 represent the recovery coefficientand the recovery exponent for ZD. In addition to elastic material constants the model has the following

inelastic material constants: DO, ZO, ZI, Ze, Z3, ml, m2, ¢xl, A2, rl, re. and n. In most cases one can set rl =

re, A 1 --A 2, and Z0 = Z2. The values of all these constants for the material B1900+Hf, taken from reference

18, are listed in table II.

NUMERICAL INTEGRATION METHODS

To develop efficient, accurate, and stable explicit integration strategies for integrating the constitutive(differential) equations of the F-V and B-P viscoplastic models, four explicit numerical integration methodswere considered. The reasons for considering only the explicit integration methods have been explained in the

introduction. To keep the presentation simple, the mathematical forms of the integration strategies have beenpresented for only one differential equation. Generalization of these integration strategies for a system ofdifferential equations is straightforward and presented subsequently.

Consider the first-order differential equation

-J = f(t,y) (32)dt

Let the solution, denoted by y, of this differential equation be known at time t. The mathematical formulas forthe four methods are given here to calculate the solution at time t+h where h denotes the increment in time t.

Runge-Kutta, Second-Order Method (or RK2 Method)

For equation (32) the approximate solution at t+h using the Runge-Kutta method is given by the well-known expressions (see, e.g., ref. 26)

y(t+h)=y+l(kl +k2) (33)

with

k l=hf(t,y); k 2=hf(t+h,y+k 1)

Lower Order Runge-Kutta Methods (RK2L and RK1L Methods)

Fehlberg (ref. 15) has proposed several lower order Runge-Kutta formulas for the solution of a systemof ordinary differential equations and has applied them to heat transfer problems. He has shown for theseproblems that the low-order, Runge-Kutta methods are considerably faster and of equal or better accuracy thanthe conventional explicit integration formulas. Higher order Runge-Kutta formulas offer no special advantagesbecause the increase in the integration step size is restricted by the stability considerations resulting from theexponential character of the solution.

It is well established that small truncation errors lead to efficient Runge-Kutta integration formulasand that the permissible integration step size is inversely related to the magnitude of these truncation errors.The attempt in the lower order integration formulas proposed by Fehlberg has, therefore, been to keep thetruncation errors as small as possible. In this study we selected two lower order Runge-Kutta methods proposedby Fehlberg: the lower Runge-Kutta method of order two (or RK2L method) and the lower Runge-Kuttamethod of order one (the Euler-Cauchy or RK1L method)). The general formula for a lower Runge-Kuttamethod of order n is given next.

Following Fehlberg (ref. 15), the integration formulas for a lower Runge-Kutta method of order n canbe written as

n

y(t + h) = y(t) + h _ c_¢f_:_=0

(34)

and

fl t+t_Kh,y +r-k_O_ fkfK = (35)

The coefficients tzr, [_rX, and cr are determined by using Taylor's expansion. The values of these coefficients

for lower order Runge-Kutta methods of orders one and two, taken from Fehlberg (ref. 15), are listed in tables

IT[ and IV. Complete derivation of these formulas and estimation of the coefficients may be found in thisreference.

Exponential Integration Method (or RK2EXP Method)

Several explicit and implicit exponential integration algorithms have been proposed from time to timefor the integration of the "stiff" constitutive equations of viscoplastic models (refs. 11 to 13). Theseexponential integration algorithms are claimed to be considerably more stable (explicit algorithms) orunconditionally stable (implicit algorithms) in the "stiff' region of integration provided that the integrandsatisfies certain mathematical requirements. Applying some implicit exponential integration algorithms tostress analysis of structural engineering problems involving viscoplastic models (ref. 27) has shown them to bemore efficient than the conventional integration methods.

Explicit integration algorithms offer several advantages for the stress analysis of structural engineeringproblems involving many degrees of freedom. Therefore, an integration strategy based on an explicit

exponential integration method (ref. 16) was developed for application to structural (and life) analysisproblems involving viscoplastic models. The exponential integration method developed in reference 16 for usein conjunction with the explicit Runge-Kutta method is shown to possess better stability characteristics thanthe conventional explicit integration methods. A built-in capability of the method automatically determineswhether the problem is stiff or nonstiff and efficiently integrates it accordingly. This exponential integrationalgorithm has successfully been applied to several mathematically stiff problems and shown to be veryefficient. A brief description and derivation of this integration method is followed by its numericalimplementation and the description of the integration strategy for application to viscoplastic models.Rigorous mathematical details are avoided to keep the presentation simple. Readers interested in completedetails may find these in reference 16.

Integrating equation (32) from t to t+h yields

t+h

y(t + h)= y(t)+ f f[s, y(s)_lS

Smt

(36)

This is the fundamental theorem of integral calculus.

To develop an exponential integration scheme, start with the Laplace integral

A

I = f e -xs F(s) ds,0

where x > 0 (37)

and F(s) is of the exponential order. Write

or

F(s) = e -ms F(s)e ms

by using Mclaurin' 12s expansion. Choose m such that

F'(O)+mF(O) =0 or

Therefore,

Integrating gives

m = -F'(O)IF(O)

A

I = fe -(x+m)sF(s) e ms ds0

F(O) Ii_e-(X+m)A ]l='_-_m L

This result is used to accomplish the exponential integration of equation (32).

(38)

(39)

(40)

(41)

(42)

10

Substituting s = t + u into equation (35) yields

y(t + h) = y(t) +

h

f e mu y'(t + u)emUdu

u=0

(43)

Also

e mUy' (t + u) = y' (t) + [y" (t) + my' (t)]u + O(u 2) (44)

Again, choose m such that

y"(t)+my'(t)=O or m=-y"(t)ly'(t) (45)

Using equations (42) to (45) gives

I - e -mh

y(t + h) = y(t) + y'(t) (46)m

For m > 0, equation (46) yields an approximation to equation (32) that is useful for a large m or a small h. For

m < 0 and decaying exponential (stiff) behavior this method is not applicable. In this case use an explicit

integration (e.g., second-order Runge-Kutta) method as the nonstiff integration method. It is important to notethat by examining the sign of the exponent m the RK2EXP method automatically determines whether the

given differential equation is stiff (m > 0) or nonstiff (m < 0) in the region of integration and integrates itaccordingly.

form:

NUMERICAL IMPLEMENTATION

The procedure for implementing the exponential integration method is given here in a step-by-step

(1) Calculate derivatives at the start of the step.

f[t,y(t)]--dold

(2) Calculatethe Euler values.

Z(t + h) = y(t) + h 4t, y(t)]

(3) Calculate the new derivatives.

f[t + h,z(t + h)] -=dne w

Note that these calculations are required also for the classical Runge-Kutta method.

If dold * O,

Calculate m = (dol d - dnew)/(h x dold)

11

Ifm >0,

Calculate the exponential approximation

y(t + h) -- y(t) + dol d x (1 - exp(-m x h)/m

EndifEndif

If dold = 0 or m < 0,

Calculate the RK2 approximation

y(t + h)= y(t)+(hJ2)X(dol d * dne w)

En_f

ESTIMATION OF LOCAL ERROR (IMBEDDING TECHNIQUE)

To estimate the local error in the solution, an "imbedding" technique is employed. In this techniquethe local error is defined as the difference in the solutions obtained by using two methods of different orders.For the RK2 method the local error is the difference between the RK2 and readily available Euler values. Forthe exponential method it is estimated by employing the exponential approximation (see ref. 16)

Ya(t + h)= y(t) exp[y'(t)hly(t)]+ O(h21 (47)

where the subscript a refers to an approximation to the integration method.

AUTOMATIC TIME-STEP INTEGRATION

Two schemes were adopted to make the integration strategy self-adaptive, that is, capable ofautomatically picking suitable time steps that preserve the stability and accuracy of the solution:

(1) Scheme I.stepas

Employed herein, scheme I defines the error in the solution during the current time

(48)

(2) Scheme II. It defines the error in the solution during the current time step as

Ir-yol

Generalization to a system of n equations is easily obtained by defining an error norm e as

(49)

e = [i=1

(50)

12

The integration strategy runs as follows:

(I) Prescribe the upper e u and lower et limits for the error norm e.

(2) If e > e u, halve the time step to hi2. (If necessary, repeat this until a value of h is obtained that

satisfies the condition e < eu.)

(3) If e < et, double the current time step to 2h.

(4) If et _- e __eu, retain the current time step.

Proceed to the next step calculations with the time step rendered by the preceding and so on.

APPLICATIONS

To illustrate the applications of the self-adaptive time integration strategy in conjunction with the fourexplicit integration methods mentioned earlier, computer programs in FORTRAN were developed andapplied to several uniaxial problems. The uniaxial problems considered consisted of tensile, relaxation,and cyclic thermal/mechanical loadings.

Figures l(a), (b), and (c) exhibit tensile, relaxation, and cyclic loadings for the F-V model. The

tensile loading (fig. l(a)) consisted of a stress rate of 6.10× 10 -3 MPa/s at 149 °C. For relaxation loading(fig. l(b)) the specimen was deformed to a strain of 0.015 in 1000 s and then allowed to relax at the same

strain for 10 000 s. For in-phase cyclic thermomechanical loading (fig. l(c)) the specimen was cycledbetween _-!-0.00375 swain in 1000 s. The temperature of the specimen was cycled (in phase with the strain)

between 200 °C (at -0.00375 strain) and 500 °C (at 0.00375 strain) in 1000 s.

The tensile, relaxation, and cyclic thermal/mechanical loadings for the B-P model are shown infigures 2(a), (b), and (c), respectively. The tensile loading (fig. 2(a)) consisted of loading the specimento 0.010 strain in 120 s at 871 °C. For relaxation loading (fig. 2(b)) the specimen was loaded to 0.008strain in approximately 100 s at 871 °C and then allowed to relax for 1000 s. For cyclic thermal/mechanical loadings (fig. 2(c)) the strain and temperature were cycled in phase between £-0.004 and 538and 760 °C, respectively. The time for one cycle was 160 s.

RESULTS AND DISCUSSION

The constitutive equations of the F-V and B-P viscoplastic models were integrated in conjunction withthe Runge-Kutta second-order method (RK2), the lower Runge-Kutta method of order one or the Euler-Cauchy (RK1L), the lower Runge-Kutta method of order two (RK2L), and the exponential (RK2EXP)integration method. The respective tensile, relaxation, and cyclic thermal/mechanical loadings wereconsidered in numerical computations for the F-V and B-P models. The results of these computations areshown in figure 3 for the F-V model and in figure 4 for the B-P model.

These computations were performed to ensure that the implementation of the self-adaptive integrationstrategy in conjunction with these four explicit integration methods yielded accurate results. Thecomputations presented in figures 3 and 4 were carried out by assigning very small error tolerances forintegration for different types of loadings (described earlier). The curves marked as "exact" in thesefigures were obtained by using the Runge-Kutta, second-order method (without self-adaptive integrationstrategy) and choosing a very small time step for integration of the constitutive differential equations ofthe two viscoplastic models. Figures 3 and 4 show that for both viscoplastic models and all uniaxialproblems (tensile, relaxation, and cyclic loadings), the self-adaptive integration strategies produced

results that are in close agreement with the "exact" results. This conclusion gives confidence in accurateimplementation of the self-adaptive integration strategies for the four explicit integration methods.

To test the efficiency of different numerical integration algorithms in conjunction with the self-adaptive time integration strategy described earlier, a large number of cases, involving the same loadingsand viscoplastic models, were run. Different values and combinations of error tolerances were consideredin numerical computations involving different types of loadings. Because it is not possible to list all the

13

results, the most significant results from these calculations for the three types of loadings are given intable V for the F-V model and in table VI for the B-P model.

A criterion must be established for comparing the efficiencies and accuracies of different integrationalgorithms. The following criterion was adopted in the present study. First, an arbitrary and desiredaccuracy (or acceptable error) in the solution was assumed. Then, the error tolerances for each integrationalgorithm and error control scheme were varied until the required accuracy was achieved in the solution.This was done for each type of loading and viscoplastic model. The execution time (CPU time) for eachcase was recorded. The execution time for tracing a given loading path when using the RK2EXP (or anyother) integration algorithm depended not only on the error tolerances but also on the scheme being usedto define the time steps during the integration. Of the execution times thus obtained, the smallest time for

each integration algorithm and the corresponding error tolerances, scheme, etc., were selected. Thesetimes are listed in tables V and VI for all loadings and viscoplastic models. These tables readily reveal the

most efficient integration algorithm for a given viscoplastic model and given loading.

It is advantageous to define the format of tables V and VI at this stage. The tables have identicalformats. Column 1 lists the integration algorithms investigated. Column 2 indicates which error norm

(scheme), equation (48) or equation (49), was used to control the time step for the self-adaptive timeintegration strategy. The prescribed upper error limits RTOLu for the Runge-Kutta methods and ETOLu for

the exponential integration method are shown in columns 3 and 4, respectively. These values represent thevalue of eu (see section Automatic Time-Step Integration) for the two integration methods. An additional

advantage of using the RK2EXP method, as may be noted from columns 3 and 4, is that it allows variouscombinations of error tolerances RTOLu and ETOLu, making the RK2EXP integration method more

versatile than the other integration methods. Column 5 shows the total number of steps (iterations)

required by a given method to trace the given loading path (cycle). The number of steps for the RK2EXPmethod in column 5 is split into two parts, a and b. The superscript a on a number denotes the number oftimes the RK2 method was used, and the superscript b denotes the number of times the exponentialmethod was used. Column 6 lists the execution times (CPU times) in seconds (on a Cray-YMP computer)taken by various integration methods to trace a given loading path. Finally, the accuracy of a givenintegration method is indicated in column 7. This column lists the error in the result obtained by using agiven integration algorithm. The error is defined as

Error = (Yc- Ye)IYe

where Yc denotes the solution obtained by using the current integration algorithm and Ye designates the

"exact" solution obtained by using the RK2 method with a constant and small time step as describedearlier.

Results Using Freed-Verrilli Model

Tensile loading.raThe tensile loading shown in figure l(a) and the F-V model were used to generate

the results listed in table V(a). The RK1L and RK2L integration algorithms were found to be less efficientthan the RK2 method. The execution times for the former two methods were 10.002 and 12.949 s, respec-

tively, whereas for the same accuracy (0.460x 10-3) the time taken by the RK2 method was 9.871 s. The

most interesting result for the tensile loading case is obtained from the fourth row of the table. It shows theRK2EXP method to be significantly more efficient (by a factor of about 2) and slightly more accurate

(0.416x 10-3) than the RK2 method (0.460x 10-3). Fewer iterations (40 352) were performed by the

RK2EXP method to trace the tensile loading path than by RK2 and the other integration methods(104 091).

Relaxation loadin2.--The results of integrating the F-V model over the relaxation loading path of

figure l(b) by the different integration methods are shown in table V(b). For the same accuracy the RK1Land RK2L methods were comparable to or less efficient than the RK2 method. The RK2EXP method, for acomparable accuracy, took less execution time (1.096 s) than the RK2 method (1.205 s). The RK2EXPmethod was about 10 percent more efficient (faster) than the RK2 method and took fewer steps (8381)

14

than the RK2 method (10 802). Using the RK2EXP method in this case is clearly advantageous as, for

comparable accuracy, it is more efficient than the RK2 method.

Cyclic loading.--Table V(c) summarizes results for the cycling loading (see fig. l(c)) obtained with

the F-V model. The RK2, RK1L, and RK2L integration methods gave almost the same accuracy (column7). However, RK1L and RK2L were not as efficient as the RK2 method. The execution times of 3.554 and

4.487 s for the RK1L and RK2L methods were higher than that for the RK2 method (3.508 s). Comparingthe results for the RK2 and RK2EXP methods (rows 1 and 5) shows that the RK2EXP method took less

execution time (3.259 s) than the RK2 method (3.508 s). However, the error was larger (0.343x 10-3) for

RK2EXP than for RK2 (0.227x 10-3). Therefore, for cyclic loading the RK2 and RK2EXP methods are

comparably advantageous and either may be used.

Results Using Bodner-Partom Model

Tensile loading.raThe results for the tensile loading of figure 2(a) and the B-P model are shown in

table VI(a). The total numbers of iterations performed using the RK2, RK1L, and RK2L integrationmethods were the same. However, the RK2EXP integration method took only 2129 steps to trace the giventensile loading path. The most important results of the tensile loading case are obtained from columns 6and 7. Column 6 shows that for the present case the RK2EXP method took significantly less executiontime (0.223 s) than the RK2 (1.166 s), RK1L (1.183 s), and RK2L (1.620 s) integration methods. Also the

RK2EXP method yielded a more accurate solution (error of 0.104x 10--3) than the RK2 method (error of

0.174×10--3). Therefore, using the RK2EXP method to integrate the constitutive equations of the B-P

model is of significant advantage in this case.

Relaxation loading.raTable VI(b) displays the results for relaxation loading (see fig. 2(b)) obtained by

the different integration methods. For a given accuracy the execution times were higher for the RK1L andRK2L integration methods than for the RK2 method. The execution time was also marginally higher forthe RK2EXP method (6.653 s) than for the RK2 method (6.270 s). The total numbers of iterationsperformed by the different integration methods to trace the same relaxation loading path were comparable.In this case the RK2EXP method was found (at best) to be comparable to the RK2 method.

Cyclic Ioading._The results of integrating the constitutive equations of the B-P model for different

integration methods are listed in table VI(c) for the cyclic loading of figure 2(c). For a given accuracythe execution times were higher for the RK1L, RK2L, and RK2EXP integration methods than for the RK2integration method. The total number of iterations performed for the RK2, RK1L, and RK2L methods werethe same (96 058). However, for the RK2EXP method, it was 96 059. The RK2EXP method does not offerany particular advantage with regard to efficiency and accuracy in this case.

CONCLUSIONS

The results presented for both viscoplastic models and three types of loadings show that, forcomparable accuracy, the lower Runge-Kutta methods of orders one and two are less efficient than theRunge-Kutta, second-order and exponential integration methods. The lower Runge-Kutta methods are,therefore, not advantageous for application to viscoplastic models. The results also show that, forcomparable accuracy, the efficiency of the integration algorithms depends on the type of loading.For example, for tensile loading, for comparable accuracy and for both viseoplastic models, theexponential integration method is much faster than the Runge-Kutta, second-order method. However, forthe relaxation and cyclic loadings and for comparable accuracy, the exponential method is at best asefficient as the Runge-Kutta, second-order method.

Because, in general, for both viscoplastic models and all the loadings considered, the exponentialintegration algorithm provides significantly higher (tensile loading) or comparable (relaxation and cyclicloadings) efficiency of any other integration algorithm, its use for integrating the "stiff" equations ofviscoplastic models may be advantageous. Note that the results presented in this paper pertain to uniaxialloadings and uniaxial forms of the viscoplastic models. Computer programs in FORTRAN were developed

15

to test the efficiency and accuracy of various integration algorithms with self-adaptive strategy. Thestructural engineering problems, however, involve multiaxial loadings and forms of the viscoplasticmodels. To judge the efficiency and accuracy of the exponential integration algorithm for application tothese problems, it should be implemented into a finite element code. The process of implementing theexponential integration algorithm with self-adaptive strategy into a finite element code is in progress. Theresults of this work will be reported subsequently.

REFERENCES

1. Orowan, E.: The Creep of Metals. J. W. Scotl. Iron Steel Inst., vol. 54, 1946, pp. 45-53.

2. Kumar, V.; Morjaria, M.; and Mukherjee, S.: Numerical Integration of Some Stiff ConstitutiveModels of Inelastic Deformation. ASME J. Eng. Matls. TechnoL, vol. 102, 1980, pp. 92-96.

3. Banthia, V.; and Mukherjee, S.: On an Improved Time Integration Scheme for Stiff ConstitutiveModels of Inelastic Deformation. ASME J. Eng. Matls. TechnoL, vol. 107, 1985, pp. 282-285.

4. Hart, E.W.: Constitutive Relations for the Nonelastic Deformation of Metals. ASME J. Eng. Marls.Technol., vol. 98, 1976, pp. 97-105.

5. Gear, C.W.: Algorithm 407 DIFSUB for Solution of Ordinary Differential Equations. Communicationsof the ACM., vol. 14, 1971, pp. 176-179.

6. Cormeau, I.: Numerical Stability in Quasi-Static Elastic/Viscoplasticity. Int. J. Numer. Methods inEng., vol. 1, 1975, pp. 109-127.

7. Hughes, T.J.R.; and Taylor, R.L.: Unconditionally Stable Algorithms for Quasi-Static Elasto/Viscoplastic Finite Element Analysis. Computers and Structures, vol. 8, 1978, pp. 169-173.

8. Hornberger, K.; and Stamm, H.: An Implicit Integration Algorithm With a Projection Method forViscoplastic Constitutive Equations. Int. J. Numer. Methods in Eng., vol. 28, 1989, pp. 2397-2421.

9. Lambert, J.D.: Computational Methods in Ordinary Differential Equations. Wiley, London, 1973.

10. Roberts, C.E. (Jr.): Ordinary Differential Equations - A Computational Approach. Prentice-Hall, 1979.

11. Walker, K.P.: Research and Development Program for Nonlinear Structural Modelling With Advanced

Time-Temperature Dependent Constitutive Relationships. NASA CR-165533, 1981.

12. Walker, K.P.: A Uniformly Valid Asymptotic Integration Algorithm for Unified Constitutive Models.Advances in Inelastic Analysis. S. Nakazawa et al., eds., ASME, New York; AMD-Vol. 28 &PED-Vol. 28, 1987, pp. 13-27.

13. Freed, A.D.; Yao, M.; and Walker, K.P.: Asymptotic Integration Algorithms for First Order ODE'sWith Application to Viscoplasticity. NASA TM-105587, 1992.

14. Arya, V.K.; Hornberger, K.; and Stature, H.: On the Numerical Integration of Viscoplastic Models.Report No. KfK--4082, Kernforschungszentrum, Karlsruhe, Germany, 1986.

15. Fehlberg, E.: Low-Order Classical Runge-Kutta Formulas With Stepsize Control and Their Applica-tion to Some Heat Transfer Problems. NASA TR R-315, 1964.

16. Ashour, S.S.; and Hanna, O.T.: Explicit Exponential Method for the Integration of Stiff OrdinaryDifferential Equations. AIAA J. Guidance, vol. 14, 1991, pp. 1234-1239.

17. Freed, A.D.; and Verrilli, M.: A Viscoplastic Theory Applied to Copper. NASA TM-100831, 1988.

16

18.Bodner,S.R.;andPartom,Y.:ConstitutiveEquations for Elastic-Viscoplastic Strain HardeningMaterials. ASME J. Appl. Mech., vol. 42, 1975, pp. 385-389.

19. Bodner, S.R.: Constitutive Equations for Dynamic Material Behavior. Mechanical Behavior ofMaterials Under Dynamic Loads, U.S. Lindholm, ed., Springer-Verlag, New York, 1968.

20. Gilman, J.J.: Dislocation Mobility in Crystals. J. Appl. Phys., vol. 36, 1964, p. 3195.

21. Johnston, W.G. and Gilman, J.J.: Dislocation Velocities, Dislocation Densities and Plastic Flow inLithium Fluoride Crystal. J. Appl. Phys., vol. 30, 1959, pp. 129-144.

22. Bodner, S.R.; and Partom, Y.: A Large Deformation Elastic-Viscoplastic Analysis of a Thick-Walled

Spherical Shell. ASME J. Appl. Mech., vol. 39, 1972, pp. 751-757.

23. Stouffer, D.C.; and Bodner, S.R.: A Constitutive Model for the Deformation Induced AnisotropicPlastic Flow of Materials. Int. J. Eng. Sci., vol. 17, 1979, pp. 757-764.

24. Bodner, S.R.; and Stouffer, D.C.: Comments on Anisotropic Plastic Flow and Incompressibility. Int. J.Eng. Sci., vol. 21, 1983, pp. 211-215.

25. Bodner, S.R.: Evolution Equations for Anisotropic Hardening and Damage of Elastic-ViscoplasticMaterials. Plasticity Today--Modelling, Methods and Applications, A. Sawczuk and G. Bianchi, eds.,Elsevier Applied Science, 1984.

26. Horneck, R.W. : Numerical Methods. Prentice-Hall, New Jersey, 1975.

27. Iskovitz, I.; Chang, T.Y.P.; and Saleeb, A.F.: Finite Element Implementation of State Variable-BasedViscoplastic models. High-Temperature Constitutive Modelling--Theory and Application, A. Freedand K. P. Walker, eds., ASME MD--Vol. 26 & AMD-Vol. 121, 1991, pp. 307-321.

TABLE I.--FREED-VERRILLI MODELCONSTANTS FOR COPPER

[From reference 17.]

Constant Unit Value

E

Ot

V

A

Doh

H

L

n

QS

r,,,

MPa

oc-1

s-I

MPa

MPa

MPa

MPa

J/mole

MPa

K

165 000 - 125 T15x10"-6 + 5x 10--9 T

0.34

50x I06

1.5

500

5000

25 exp(- T/300)5

200 000

14.3

1356

17

TABLErl.--BODNER-PARTOM MODEL CONSTANTS FOR B 1900+Hf

[From reference 18.]

(a) Temperature-independent constants

m I = 0.270 MPa-1

m 2 = 1.520MPa-1

(X 1 =0.0

Z I = 3000 MPa

23 = 1150 MPa

r ! =r 2 =2

Do = lx104 s -I

(b) Temperature-dependent constants

Constant

n

ZO (MPa)

Al =A2 (s -l)

Z2 -- Zo (MPa)

_760

1.0552700

0

2700

Temperature, °C

871 982

1.03 0.85002400 1900

0.0055 0.0200

2400 1900

1093

0.70001200

0.2500

1200

(c) Elastic constants (MPa; T in °C)

E= 1.987x 105 + 16.78 T- 0.1034 T2 + 1.143x 10-5 T 3

G = 8.650x 104 - 17.58 T + 0.2321 T 2 - 3.464x 10 -5 T 3

TABLE Ill.--COEFFICIENTS FOR LOWER RUNGE-KU'I_A

METHOD OF ORDER ONE (EULER-CAUCHY

OR RK1L METHOD)

K

0

1

(X K

0.0

.5333333

(X=o)

0.5333333

CK

0.0625000

.5333333

TABLE IV.---COEFFICIENTS FOR LOWER RUNGE-KUTTA

METHOD OF ORDER TWO (RK2L METHOD)

K Otr

0 0.0

1 .2500000

2 .6750000

_1¢X

0.0

.2500000

-.2362500 0.9112500

C K

0.2401795

.0303030

.7295174

18

Method

RK2RK1LRK2LRK2EXP

Scheme

III

II

TABLE V._RESULTS FOR FREED-VERRILLI MODEL

(a) Tensile loading

Prescribed

upper error inRunge-Kutta

method,RTOLu

10--3

10-3

10-3

10-7

Prescribed

upper error inexponential

method,

ETOLu

5x 10- 6

Iterations

104091

104091

104091

40 352 (3474a+36 878 b)

(b) Relaxation loading

Execution

(CPU) time,s

9.871

10.002

12.949

4.638

Error

0.460x 10- 3

.460x 10- 3

.460x 10- 3

.416×10- 3

RK2

RK1L

RK2L

RK2EXP

I

I

I

lI

10-2

10- 2

10- 2

10- 4 10- 4

10 802

10 802

10 802

8381 (1001a+7380b)

1.205

1.205

1.530

1.096

0.352x 10-3

.352xl 0-3

.352x 10- 3

.364x 10- 3

(c) Cyclic loading

RK2

RK1L

RK2L

RK2EXP

I

I

I

lI

10-2

10-2

10-2

10-6 10- 4

aNumber of times Runge-Kutta method was used.

bNumber of times exponential method was used.

32 105

32 102

32 105

26 160 (11 157a+15003 b)

3.508

3.554

4.487

3.259

0.227x 10- 3

.223x10-3

.223x 10-3

.343x 10--3

19

TABLEVI.--RESULTSFORBODNER-PARTOM MODEL

(a) Tensile loading

Method Scheme

RK2 I

RK1L I

RK2L I

RK2EXP lI

Prescribedupper error inRunge-Kutta

method,RTOLu

10-2

10-2

10-2

10-1

Prescribedupper error inexponential

method,ETOLu

10-1

(b) Relaxation loading

Iterations

11 510

11 510

11 510

2129 (1948a+181 b)

Execution(CPU) time,

s

1.166

1.183

1.620

.223

Error

0.174x10-3

.174x10-3

• 174x 10-3

.104x 10-3

RK2

RK1L

RK2L

RK2EXP

I

I

I

lI

10-3

10-3

10-3

10-3 10-3

(c) Cyclic loading

61 000

61 000

61 000

61 008 (50 107a+10 901b)

6.270

6.299

8.712

6.653

0.525x 10-4

.525x 10- 4

.525x 10- 4

.525x 10- 4

RK2

RK1L

RK2L

RK2EXP

I

I

I

H

10-5

10-5

10-5

10-5 10-5

96 058

96 058

96 058

96 059 (62 941a+33 l18b)

10.117

10.362

13.966

10.971

0.858x 10-5

.858x 10-5

.858x10-5

.858x10-5

aNumber of times Runge-Kutta method was used.

bNumber of times exponential method was used.

20

.0oel

¢oto

==

(a)

.015

.010

.005

0

I I I I -.oo5 _, I10 20 30 40x104 0 1000 _ 11 000

Time, s Time, s

._c

oo

GO.

E

500 .00375

35O_c

o(/)

50O 1000

(c) I200 -.00375

0 1500 0

Time, s

50O 1000

Figure 1.--Thermal/mechanical Ioadings for Freed-Verrilli model. (a) Tensile loading. Temperature, 149 °C. (b) Relaxation loading.

Temperature, 316 °C. (c) Cyclic loading. Temperature, 200 <=>500 °C.

I1500

21

.010

I I0 40 80 120

Time, s

r.

w

c-e3r-E)

:Z

,0O8

I100

Time, s

r_ i1100

2

E

649

76O

(c538 I

0 80 160 240 0 80 160 240Tlme, s

•_. 0

F_ure 2.--Thermallmechanical loadings for Bodner-Partom model. (a) Tensile loading. Temperature, 871 °C. (b) Relaxation loading.

Temperature, 871 0(3. (c) Cyclic loading. Temperature, 538 <=>760 °C.

22

200 -

1000 --

150

.=:S

_1oo

u)

5o

4O

:S

@3

.eu)

2o

60 --

fo[]

RK2RK1 LRK2LRK2EXPExact

(a) I I I I I

.05 .10 .15 _o0 _25Strmn

(b) I I I I•005 .010 .015 .020

Strain

o.:E

e_

03

75O

5OO

250 ¸

#f Z_ RK1L/ 1-1 RK2L

• RK2EXP

- Exactl I : :ct (a)l

.002 .004 .006 .008 .010Strain

1000

0 .... 008Strain"

IOO0

.010

80 -

5O0

4O

#

:_ _ o

o

-8O (c) I I I I

--.004 -.002 0 .002 .004Strain

Figure 3.--_tress-strain and relaxation curves andhysteresis loops for Freed-Vernlli model (copper).(a) Stress-strain curve. Stress rate, 6.10xl 0-3 MPa/s;temperature, 149 °C. (b) Relaxation curve. Strain

rate, 1.50xl 0-6/s; temperature, 316 °C. (c) Hysteresisloops. Strain rate, 1.50xl 0-5/s; strain range, _+0.00375;

temperature, 200 <_ 500 °C.

-1000 I I I (c)I--.004 --.002 0 .002 .004

Strmn

Figure 4._tress-strain and relaxation curves and hysteresisloops for Bodner-Partom model (B1900+Hf). (a) Stress-strain

curve. Strain rate, 8.333x10-6/s; temperature, 871 °C.{b) Relaxation curve. Strain rate, 8.333x10-6/s; maximumstrain, 0.008; temperature, 871 °C. (c) Hysteresis loops. Strain

rate, 1.0xl 0-4/s; strain range, !-0.004; temperature, 538 <=>760 °C.

23

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1. AGENCY USE ONLY (Leave _ank-) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

August 1994 Final Contractor Report

4. TITLE AND SUBTITLE

Efficient and Accurate Explicit Integration Algorithms With Application to

Viscoplastic Models

6. AUTHOR(S)

V]nod K. Arya

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

5. FUNDING NUMBERS

WU-505-63-5B

NCC3-120

8. PERFORMING ORGANIZATIONREPORT NUMBER

University of ToledoToledo, Ohio 43606

9. SPONSORING/MONITORINGAGENCYNAME(S)ANDADDRESS(ES)

National Aeronautics and Space AdministrationLewis Research Center

Cleveland, Ohio 44135-3191

E-8930

10. SPONSORING/MONITORINGAGENCY REPORT NUMBER

NASA CR-195342

11. SUPPLEMENTARY NOTES

Project Manager, Gary R. Halford, Structures Division, NASA Lewis Research Center, organization code 5200,

(216) 433--3265.

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13. ABSTRACT (Maximum 200 words)

Several explicit integration algorithms with self-adaptive time integration strategies are developed and investigated for

efficiency and accuracy. These algorithms involve the Runge-Kutta second order, the lower Runge-Kutta method oforders one and two, and the exponential integration method. The algorithms are applied to viscoplastic models put forth

by Freed and Verdlli and Bodner and Partom for thermal/mechanical loadings (including tensile, relaxation, and cyclic

loadings). The large amount of computations performed showed that, for comparable accuracy, the efficiency of an

integration algorithm depends significantly on the type of application (loading). However, in general, for the aforemen-tioned loadings and viscoplastic models, the exponential integration algorithm with the proposed self-adaptive time

integration strategy worked more (or comparably) efficiently and accurately than the other integration algorithms. Using

this strategy for integrating viscoplastic models may lead to considerable saving in computer time (better efficiency)without adversely affecting the accuracy of the results. This conclusion should encourage the utilization of viscoplastic

models in the stress analysis and design of structural components.

14. SUBJECT TERMS

Integration algodthras; Numerical methods; Viscoplasticity; "Stiff' differential

equations

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